Theorem termco 50259

Description: The object of a terminal category. (Contributed by Zhi Wang, 17-Nov-2025.)

Hypotheses

Ref	Expression
termcbas.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcbas.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
termco	⊢ (𝜑 → ∪ 𝐵 ∈ 𝐵)

Proof of Theorem termco

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		termcbas.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2		termcbas.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3	1, 2	termcbas 50258	. 2 ⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})
4		unieq 4883	. . . . . 6 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 = ∪ {𝑥})
5		unisnv 4892	. . . . . 6 ⊢ ∪ {𝑥} = 𝑥
6	4, 5	eqtrdi 2814	. . . . 5 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 = 𝑥)
7		vsnid 4629	. . . . 5 ⊢ 𝑥 ∈ {𝑥}
8	6, 7	eqeltrdi 2871	. . . 4 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 ∈ {𝑥})
9		id 23	. . . 4 ⊢ (𝐵 = {𝑥} → 𝐵 = {𝑥})
10	8, 9	eleqtrrd 2866	. . 3 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 ∈ 𝐵)
11	10	exlimiv 1960	. 2 ⊢ (∃𝑥 𝐵 = {𝑥} → ∪ 𝐵 ∈ 𝐵)
12	3, 11	syl 18	1 ⊢ (𝜑 → ∪ 𝐵 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {csn 4589  ∪ cuni 4872  ‘cfv 6536  Basecbs 17264  TermCatctermc 50250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-termc 50251

This theorem is referenced by: (None)